SOLUTIONS FOR HOMEWORK 6
13.1. (b) We have to show that, for any > 0, there exists > 0 with
the property that n |f (bk )g(bk ) f (ak )g(ak )| < whenever ([ak , bk ])n are
k=1
k=1
non-overlapping intervals with n (bk ak ) < . Note that the functions f
k=1
HOMEWORK 2
Problems assigned up to and including Friday, 04/08. The solutions
are due on Friday, 04/15.
Assigned Monday, 04/04:
Chapter 17: 17.4, 17.5, 17.6, 17.7, 17.12, 17.13.
Note. In Exercise 17.13, you can use the fact that any measure space
can be s
HOMEWORK 4
Problems assigned up to and including Friday, 04/29. The solutions
are due on Friday, 05/13.
Assigned Wednesday, 04/27:
Chapter 23: 23.2, 23.4, 23.12, 23.13, 23.14.
Back to the syllabus.
Back to the main page of the course.
The solutions.
1
SOLUTIONS FOR HOMEWORK 3
17.8. Note that (a) follows from (b). Indeed, dene the measure via (E) =
(E)/(X). Then Mp (f ) = f Lp () . So, we prove (b). Throughout the rest of the
proof, (X) (0, 1]. The proof of Theorem 17.4 shows that, in this case, f p f q
HOMEWORK 5
Problems assigned up to and including Friday, 05/13. The solutions
are due on Friday, 05/20.
Assigned Friday, 05/06:
Suppose X is a Banach space, equipped with the operation of multiplication. That is, for x, y X, we have xy X. Throughout, we a
HOMEWORK 3
Problems assigned up to and including Friday, 04/15. The solutions are due on Friday,
04/29.
Assigned Wednesday, 04/13:
Chapter 17: 17.8.
A. Suppose X is a Banach space, and X is separable. Prove that X is also separable.
Note. The dual of a se
SOLUTIONS FOR HOMEWORK 4
23.2. Solution 1. For N N, let GN = cfw_(x, y) : y = f (x), N < x N. It suces
to show that, for any N, GN belongs to (B B), and has measure 0. Henceforth, we x
N. Our goal is to produce a sequence Borel sets En (n N) so that E1 E2
SOLUTIONS FOR HOMEWORK 5
Suppose X is a Banach space, equipped with the operation of multiplication. That is,
for x, y X, we have xy X. Throughout, we assume that multiplication is linear
that is, (1 x1 + 2 x2 )y = 1 x1 y + 2 x2 y, and x(1 y1 + 2 y2 ) =
TAKE-HOME FINAL FOR MATH 210C (SPRING 2011)
The solutions are due by 2 pm on Friday, June 3. You can either give your
papers to me directly, slide them under my door, or place them in my mailbox.
I prefer to receive hard copies. If you plan to submit your
SOME PRACTICE PROBLEMS FOR THE QUALIFIER
Chapter 9: 9.14, 9.15, 9.18 9.39.
Chapter 10: 10.8.
Chapter 11: 11.4 (under the condition that (cfw_x X : f (x) 1/n) < for any n N).
Chapter 12: 12.11, 12.13.
Chapter 13: 13.9.
2
Chapter 17: 17.9 (Hint: use Hlders
SOLUTIONS FOR THE FINAL
1 (10 points): Prove Clarksons Inequality for 2 p < : if f, g Lp (), then
f +g
2
p
f g
2
+
p
p
f
p
p
p
+ g
2
p
p
.
You can assume that the functions are real-valued (the inequality also holds for complexvalued functions).
Hint. You
SOLUTIONS FOR MIDTERM
Throughout, all measure spaces are assumed to be -nite. All functions are extended real
valued.
1 (10 points): Prove the uniqueness of the weak limit. More precisely, suppose 1 p ,
(X, A, ) is a -nite measure space, and f, f1 , f2 ,
TAKE-HOME MIDTERM FOR MATH 210C (SPRING 2011)
The solutions are due by 2 pm (that is, by the beginning of the lecture) on
Friday, May 6. You can either give your papers to me directly, slide them
under my door, or place them in my mailbox. I prefer to rec
SOLUTIONS FOR HOMEWORK 1
16.3. It suces to consider the case of f 0. Let F = f r , G = f r , H = f (1)r ,
a = p/(r), and b = q/(r(1 ). Then 1/a + 1/b = 1, and F = GH. By Hlder
o
1/a
1/b
r
Inequality, F 1 G a H b . However, F 1 = f r ,
G
1/b
1/a
a
r/p
fp
=
HOMEWORK 1
Problems assigned up to and including Friday, 04/01. The solutions
are due on Friday, 04/08.
Assigned Wednesday, 03/30:
Chapter 16: 16.3, 16.4, 16.6, 16.16, 16.17, 16.22.
Note. In Exercise 16.16, you can assume that f has no atoms. Furthermore,
SOLUTIONS FOR HOMEWORK 5
b
b
12.2. We show that Va (1/f ) Va (f )/c2 . To this end, it suces to show that,
b
b
for any partition P = cfw_a = x0 < x1 < . . . < xn = b, Va (1/f, P) Va (f )/c2.
But
n
b
Va (1/f, P)
1
1
=
f (xk1 f (xk
=
k=1
1
2
c
n
k=1
n
k=1
HOMEWORK 5
Problems assigned up to and including Friday, 02/11. The solutions
are due on Friday, 02/18.
Assigned Monday, 02/07:
Chapter 12: 12.2, 12.6, 12.10, 12.12, 12.14.
Assigned Wednesday, 02/11:
A. Suppose f : [a, b] R is increasing, and a set E [a,
HOMEWORK 6
Problems assigned up to and including Friday, 02/25. The solutions
are due on Friday, 03/04.
Assigned Wednesday, 02/16:
Chapter 13: 13.1(b), 13.3, 13.4, 13.7, 13.8, 13.10, 13.12, 13.13(a,c),
13.17, 13.19.
Back to the syllabus.
Back to the main
HOMEWORK 7
Problems assigned up to and including Friday, 03/04. The solutions
are due on Friday, 03/11.
Assigned Monday, 02/28:
A. Prove that any two norms on Rn are equivalent.
Hint. Prove that any norm on Rn is equivalent to 1 , introduced in
Section 15
PRACTICE PROBLEMS FOR THE MIDTERM: THE
SOLUTIONS
The in-class midterm exam will be given on Wednesday, Feb. 2. The
material from Homeworks 1-3 (Chapters 8, 9, and 10) will be covered.
8.10. Let D0 = D. For n N, let En = Dn \Dn1 . Furthermore, let
E = capn
SOLUTIONS FOR HOMEWORK 7
A. Prove that any two norms on Rn are equivalent.
Hint. Prove that any norm on Rn is equivalent to
x = (x1 , . . . , xn ), x 1 = n |xi |).
i=1
1,
introduced in Section 15.II (for
The equivalence of norms is known to be an equival
SOLUTIONS FOR MIDTERM
1 (10 points): Suppose M > 0, (X, A, ) is a measure space, D A, and f is
an extended real valued function on D, such that
|f |p d
D
p (0, ). Show that |f | M almost everywhere on D.
1/p
M for any
Hint. What can you say about (En ),
SOLUTIONS FOR FINAL
1 (10 points): Suppose E is a subset of R of positive measure. Prove that E + E =
cfw_x + y : x, y E contains a non-trivial interval.
Hint. By Problem 13.19, there exists an interval I s.t. L (E I) > 5(I)/6. By translation, we can assu
TAKE-HOME FINAL FOR MATH 210B (MARCH 2011)
The solutions are due by 1:30 pm on Friday, March 18. You can either give your
papers to me directly, slide them under my door, or place them in my mailbox. I
prefer to receive hard copies. If you plan to submit
SOLUTIONS FOR HOMEWORK 1
8.1. It suces to show that, for any non-negative simple function
on D, satisfying D d = +, we have sup D d = + (the
supremum runs over all -integrable simple functions , satisfying 0
. By the -niteness, there exist sets of nite
HOMEWORK 1
Problems assigned up to and including Friday, 01/07. The solutions
are due on Friday, 01/14.
Assigned Wednesday, 01/05:
Chapter 8: 8.1, 8.3, 8.4, 8.5, 8.6, 8.8, 8.9, 8.11.
Back to the syllabus.
Back to the main page of the course.
The solutions
HOMEWORK 2
Problems assigned up to and including Friday, 01/14. The solutions
are due on Friday, 01/21.
Assigned Wednesday, 01/12:
Chapter 9: 9.7, 9.9, 9.10, 9.11, 9.12, 9.17, 9.19, 9.22, 9.34.
You can use the conclusion of Exercise 9.17 to solve Exercise
SOLUTIONS FOR HOMEWORK 2
9.7. (a) Let fn = fAn = f 1An . Then |fn | |f |. Moreover,
|fn | d. By Monotone Convergence Theorem,
X
|f | d =
|f | d < .
|f | d = lim
n
X
An
An
By Observation 9.2, f is -integrable on D.
(b) Let hn = fBn = f 1Bn . Then limn hn =
SOLUTIONS FOR HOMEWORK 3
A. Prove that, for any L -integrable function f on R, and any > 0, there
N
exists a step function g =
i=1 ai 1Ii (Ii are nite intervals, ai R) s.t.
|f g| dL < .
R
Hint. First suppose f is a simple function. Recall Theorem 3.25.
Su
HOMEWORK 4
Problems assigned up to and including Friday, 01/28. The solutions
are due on Friday, 02/11.
Assigned Wednesday, 01/26:
Chapter 11: 11.1, 11.2(a,b), 11.3, 11.6, 11.7.
Back to the syllabus.
Back to the main page of the course.
The solutions.
1